Extrema and Zeros in SHM

In simple harmonic motion, each kinematic quantity reaches its extreme values at specific positions.

The fundamental equations are:

$$x(t) = \pm A\cos(\omega t) \qquad \text{or} \qquad \pm A\sin(\omega t)$$ $$v(t) = \mp A\omega\sin(\omega t)\qquad \text{or} \qquad \pm A\omega\cos(\omega t)$$ $$a(t) = -\omega^2 x(t)$$

At equilibrium ($x = 0$): Displacement is zero. From $a = -\omega^2 x$, acceleration is also zero. Velocity reaches its maximum magnitude:

$$|v|_{\max} = A\omega$$

At the turning points ($x = \pm A$): Displacement is at its extreme. Velocity is zero because the object momentarily stops. Acceleration reaches its maximum magnitude:

$$|a|_{\max} = \omega^2 A$$

The direction of acceleration always points toward equilibrium.

These results follow directly from $a = -\omega^2 x$. When $|x|$ is largest, $|a|$ is largest. When $x = 0$, then $a = 0$. Velocity and displacement are complementary: velocity is greatest where displacement is zero.